A022354 -id:A022354 - cp's OEIS frontend

A208780 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 0 1 horizontally and 0 1 1 and 1 1 1 vertically.

2, 4, 4, 6, 16, 6, 10, 36, 36, 10, 16, 100, 36, 100, 16, 26, 256, 60, 60, 256, 26, 42, 676, 96, 100, 96, 676, 42, 68, 1764, 156, 160, 160, 156, 1764, 68, 110, 4624, 252, 260, 256, 260, 252, 4624, 110, 178, 12100, 408, 420, 416, 416, 420, 408, 12100, 178, 288, 31684, 660

Offset: 1

R. H. Hardin Mar 01 2012

Table starts
..2....4...6..10...16...26...42...68...110...178...288....466....754....1220
..4...16..36.100..256..676.1764.4624.12100.31684.82944.217156.568516.1488400
..6...36..36..60...96..156..252..408...660..1068..1728...2796...4524....7320
.10..100..60.100..160..260..420..680..1100..1780..2880...4660...7540...12200
.16..256..96.160..256..416..672.1088..1760..2848..4608...7456..12064...19520
.26..676.156.260..416..676.1092.1768..2860..4628..7488..12116..19604...31720
.42.1764.252.420..672.1092.1764.2856..4620..7476.12096..19572..31668...51240
.68.4624.408.680.1088.1768.2856.4624..7480.12104.19584..31688..51272...82960

			Some solutions for n=4 k=3
..1..0..0....0..1..1....1..1..0....0..1..0....0..1..1....0..1..0....0..1..1
..1..0..1....0..1..0....1..0..0....1..0..0....0..1..1....0..1..1....1..0..1
..0..1..0....1..0..1....0..1..0....0..1..1....1..0..0....1..0..0....0..1..0
..1..0..1....0..1..0....1..0..1....1..0..0....0..1..1....0..1..0....1..0..0

R. H. Hardin, Table of n, a(n) for n = 1..10018

Column 1 is A006355(n+2)
Column 2 is A206981
Diagonal is A206981 and column 2 for n>1
Column 3 is A022346(n+1) for n>2
Column 4 is A022354(n+1) for n>2
Column 5 is A022366(n+1) for n>2

Empirical for column k:
k=1: a(n) = a(n-1) +a(n-2)
k=2: a(n) = 2*a(n-1) +2*a(n-2) -a(n-3)
k=3: a(n) = a(n-1) +a(n-2) for n>4
k=4: a(n) = a(n-1) +a(n-2) for n>4
k=5: a(n) = a(n-1) +a(n-2) for n>4
k=6: a(n) = a(n-1) +a(n-2) for n>4
k=7: a(n) = a(n-1) +a(n-2) for n>4

A294116 Fibonacci sequence beginning 2, 21.

Original entry on oeis.org

2, 21, 23, 44, 67, 111, 178, 289, 467, 756, 1223, 1979, 3202, 5181, 8383, 13564, 21947, 35511, 57458, 92969, 150427, 243396, 393823, 637219, 1031042, 1668261, 2699303, 4367564, 7066867, 11434431, 18501298, 29935729, 48437027, 78372756, 126809783, 205182539, 331992322, 537174861

Offset: 0

Bruno Berselli, Oct 23 2017

Steven Vajda, Fibonacci and Lucas Numbers, and the Golden Section: Theory and Applications, Dover Publications (2008), page 24 (formula 8).

Colin Barker, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (1,1).

Cf. A000032, A000045.
Subsequence of A047201, A047592, A113763.
Sequences of the type g(2,k;n): A118658 (k=0), A000032 (k=1), 2*A000045 (k=2,4), A020695 (k=3), A001060 (k=5), A022112 (k=6), A022113 (k=7), A294157 (k=8), A022114 (k=9), A022367 (k=10), A022115 (k=11), A022368 (k=12), A022116 (k=13), A022369 (k=14), A022117 (k=15), A022370 (k=16), A022118 (k=17), A022371 (k=18), A022119 (k=19), A022372 (k=20), this sequence (k=21), A022373 (k=22); A022374 (k=24); A022375 (k=26); A022376 (k=28), A190994 (k=29), A022377 (k=30); A022378 (k=32).

a0:=2; a1:=21; [GeneralizedFibonacciNumber(a0, a1, n): n in [0..40]];

Mathematica
```
LinearRecurrence[{1, 1}, {2, 21}, 40]
```

Vec((2 + 19*x)/(1 - x - x^2) + O(x^40)) \\ Colin Barker, Oct 25 2017

a = BinaryRecurrenceSequence(1, 1, 2, 21)
print([a(n) for n in range(38)]) # Peter Luschny, Oct 25 2017

G.f.: (2 + 19*x)/(1 - x - x^2).
a(n) = a(n-1) + a(n-2).
Let g(r,s;n) be the n-th generalized Fibonacci number with initial values r, s. We have:
a(n) =     Lucas(n) + g(0,20;n), see A022354;
a(n) = Fibonacci(n) + g(2,20;n), see A022372;
a(n) =  2*g(1,21;n) - g(0,21;n);
a(n) =     g(1,k;n) + g(1,21-k;n) for all k in Z.
a(h+k) = a(h)*Fibonacci(k-1) + a(h+1)*Fibonacci(k) for all h, k in Z (see S. Vajda in References section). For h=0 and k=n:
a(n) = 2*Fibonacci(n-1) + 21*Fibonacci(n).
Sum_{j=0..n} a(j) = a(n+2) - 21.
a(n) = (2^(-n)*((1-sqrt(5))^n*(-20+sqrt(5)) + (1+sqrt(5))^n*(20+sqrt(5)))) / sqrt(5). - Colin Barker, Oct 25 2017

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A208780 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 0 1 horizontally and 0 1 1 and 1 1 1 vertically.

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A294116 Fibonacci sequence beginning 2, 21.

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